Mixed state behavior of Hermitian and non-Hermitian topological models with extended couplings

Geometric phase is an important tool to define the topology of the Hermitian and non-Hermitian systems. Besides, the range of coupling plays an important role in realizing higher topological indices and transition among them. With a motivation to understand the geometric phases for mixed states, we discuss finite temperature analysis of Hermitian and non-Hermitian topological models with extended range of couplings. To understand the geometric phases for the mixed states, we use Uhlmann phase and discuss the merit-limitation with respect extended range couplings. We extend the finite temperature analysis to non-Hermitian models and define topological invariant for different ranges of coupling. We include the non-Hermitian skin effect, and provide the derivation of topological invariant in the generalized Brillouin zone and their mixed state behavior also. We also adopt mixed geometric phases through interferometric approach, and discuss the geometric phases of extended-range (Hermitian and non-Hermitian) models at finite temperature.

www.nature.com/scientificreports/ Topological phases are characterized based on the numbers called topological invariants, which are in correspondence with the number of localized edge modes 40,41 . It is possible to generate higher winding numbers (WNs) either by increasing the number of coupling sites (static method) 42 or by periodic driving (dynamical method) 43 . However, this observation is true for BDI and AIII symmetry classes and holds good only for certain limit. The breaking of symmetry may violate the above property 44 . With the infinite neighbor coupling, one can obtain long-range models which are interesting from the perspective of massive edge modes and breaking of Lorentz invariance [44][45][46][47][48][49][50] . The long-range models have been experimentally realized in trapped ions [51][52][53][54] , atom coupled to multi-mode cavities 55 , magnetic impurities 56,57 and simulated circuits 58 . They have a advantage of suppressing the finite sized effect over to short-range models 59 . The massive edge modes are found to be an effective qubits in topological computations 48 . On the other hand, non-Hermitian system exhibit sensitivity towards the boundary conditions, and there exists non-Hermitian skin effect which creates an extra localization of eigen states in the open boundary condition.
Motivation Dealing the topological state of matter at finite temperature as a condition of mixed state is an interesting area from the perspective of both theory and experimentation. Here our motivation is three-folded.
• So far, there are many efforts to understand the behavior of geometric phase for mixed states through different approaches. In the work Ref. 23 27 ), gives the signature of analyzing the possible higher WNs through the mixed state approach (in a four band model). The authors of Refs. 31,60 shows the limitations in Uhlmann phase, and explore interferometric phase as a better geometric phase at finite temperature. However, the question about measurement of higher WNs at finite temperature still remains unanswered. In a recent work, we have observed the staircase of topological transitions in 1D extended-range models 42 . The extended-range of coupling creates higher WNs, and the model reduces to short-range with the increase in the decay parameter. The staircase of transition occurs between even-even (odd-odd) WNs for even (odd) number of interacting neighbors. Here, we are motivated to understand the possibility to define higher WNs and transitions among them at finite temperature. • Mixed state approach can be an efficient tool to understand the geometry of non-Hermitian systems, where the system is connected to a thermal bath. Here we consider the non-Hermitian model (which exhibits local imbalance in the hopping amplitudes) and try to explain the topological invariant at finite temperature. Here we also extend our interest to understand the interplay of topology and long-range effects at finite temperature. • Some topological systems exhibit localization at criticality 40,[61][62][63][64][65][66][67][68][69][70][71] , and hence there occurs a necessity of defining topology at gapless phases. In our work (Refs. 72,73 ), we have observed a transition among critical regions across a multi-critical point. This is an interesting behavior which can be observed in models extendedrange coupling. This finding signals that, bulk gap is not a necessary condition to perform bulk-boundary correspondence 40,62 . However, here we extend our studies towards topological invariant for mixed states at gapless condition and thereby to understand the relation between localized edge modes and topology of criticality at finite temperature.
Here we consider a topological model connected to a thermal bath at temperature T and can be effectively expressed through Gibb's ensemble. In order to explain the topological invariant at finite temperature, we use two different approaches called Uhlmann and interferometric phases.

Model Hamiltonian and properties
We consider a generalized two band model for a one dimensional system, which can reflect both Hermitian and non-Hermitian properties based on the system parameter 74 .
where χ x,y,z are the pseudo-spin vectors as a function of quasi-momentum k and σ x,y,z are the Pauli spin matrices. Under the Hermitian condition, the above Hamiltonian behaves H(k) = H(k) † and under non-Hermitian condition, it is H(k) = H(k) † . Because of the Hermiticity, the left and right eigenstate are defined as, with E ± (k) = ± (χ x (k)) 2 + (χ y (k)) 2 + (χ z (k)) 2 . For the Hermitian condition, we work on the orthonormal basis, i.e.,|ψ k± � = |ϕ k± � and E ± (k) = E * ± (k) . For the non-Hermitian systems, we consider biorthonormal basis, i.e., �ψ kν |ϕ kµ � = δ ν,µ and |ϕ kµ ��ψ kµ | = 1 . Here ν and µ represent the state '+ 'and '-'respectively. By normalizing with factor �ψ k,± |ϕ k,± � , we get with T as transpose operator. The geometric phase due to the adiabatic evolution is given by, www.nature.com/scientificreports/ which yield the quantized values for the gapped topological phases 19 . For Hermitian systems, the azimuthal angle ( φ ) is real and for non-Hermitian it is complex. This behavior is because of the pseudo-spin vectors, which are real for Hermitian systems and complex (at least one component) for non-Hermitian system respectively 75 , i.e., φ = φ real (k) + φ imaginary (k) . Here the real and imaginary parts contribute to the argument and amplitude respectively. The term φ imaginary (k) do not creates any effect on the topology of the system. The non-Hermiticity creates two exceptional points instead of single Dirac (gap closing) point. The curvature function around the exceptional points is given by F1(k, M) and F2(k, M) respectively and the winding number is given by 74,75 , respectively. (For the detailed derivation, refer the "Method" section). As our model contains imaginary component only in χ y term, the exceptional points are located at (0, χ im y ) and (0, −χ im y ) under the criticality condition χ 2 x + χ 2 y = 0 . Due to the non-Hermiticity, each exceptional point induces its own origin of pseudo-spin space and corresponding winding vectors (WVs). Thus we work on extended winding vectors (EWVs) to understand the non-Hermitian effect in the parameter space. i.e., which correspond to parameter space of F 1 (k, M) and F 2 (k, M) respectively (For detailed study, please refer "Method" section).
Here we consider 1D Su-Schrieffer-Heeger (SSH) model in both Hermitian 76 and non-Hermitian versions 74,75,77 . The model has its importance in understanding of different experimental realizations related to topology [78][79][80] . The model can be expressed with different range of coupling neighbors, without altering the symmetry of the system 44,50 . Based on the range of coupling, the model can be categorized as short-range, extendedrange and long-range models.
Geometry at finite temperature. The advances in the experimental realizations of the topological matter has made a significant effort in defining the topology at finite temperature. In this regard, there are efforts in defining the topological invariant through Uhlmann and interferometric phases (mixed states) as equivalent tool to Berry phase (pure state) 31 in the finite temperature limit ( T → 0 ). Here we study the possibility of defining extended-range systems at finite temperature.
When a system is in contact with the thermal bath at temperature T, it can be defined by the Gibb's state as 31 , with � k = 2E(k, M) , which can be defined over the Bloch sphere using the polar coordinates r = χ 2 y + χ 2 z and φ = tan −1 χ y (k) χ x (k) . As the system is temperature dependent, the Bloch sphere is mapped to a sphere centered at the maximally mixed state. For the topological condition, this curve makes a closed loop and for non-topological the curve confines to a single side of the maximally mixed states respectively.
In case of Hermitian models, the method is straightforward, where the Gibbs's state corresponds to a single parameter space F(k, M) . In case of non-Hermitian models, the Gibb's state can be expressed in terms of parameter spaces F1(k, M), F2(k, M) corresponding to individual exceptional points. Thus the effective geometry of the parameter space can be understood by the combined effect of F1(k, M) and F2(k, M).
Uhlmann geometric phase Here, the pure states in the Hilbert space H A H B form a total space of fiber bundles over the mixed states of H A . The geometric phase can be achieved from parallelism condition over the base manifold such that every infinitesimal change δk from the state ψ(k) † to ψ(k + δk) which equals the fidelity of density matrix [20][21][22] where ρ(k) = ψ(k)ψ(k) † and has a gauge U(N) freedom such that ρ → ψ(k)U(N)U † (k)ψ † (k) remains unchanged. With the limit T → 0 , the system drives towards pure state limit and the geometric phase becomes with Uhlmann phase 31,60 (4) γ = �ψ k,+ |i∂ k |ϕ k,+ � �ψ k,+ |ϕ k,+ � dk. www.nature.com/scientificreports/ Thus the Uhlmann geometric phase in pure state limit is technically equivalent to the Berry phase (For detailed study, please refer "Method" section). This methodology can be generalized to both Hermitian and non-Hermitian systems.
For the Hermitian systems the method is straightforward where we have single closing point and the parameter space corresponding to F(k, M) gives the information of geometric phases. The method is slightly modified in the non-Hermitian systems, where we obtain two exceptional points. Thus, the Uhlmann phase can be calculated as the average of individual geometric phases corresponding to parameter spaces of F1(k, M) and F2(k, M) , i.e., W U = W U1 +W U2 2 . Hermitian Su-Schrieffer-Heeger chain. Here we consider 1D Su-Schrieffer-Heeger chain with extended-range of intercell hopping, i.e., where t is the intracell hopping and t ′ is the intercell hopping with power law decay. The term l represents the site index with α as the decay parameter. With the limit l → ∞ (infinite neighbors) the model becomes long-range and as α → ∞ the model reduces to short-range (where the system contains only W = 1 and W = 0 phases). This model resembles the physics of another famous model Kitaev chain in the isotropic limit, where the hopping parameter (J) and pairing parameter ( ) decays with same manner J l α = � l α 42,44 . The schematic representation is given in Fig. 1a. After Fourier transformation, we write BdG Hamiltonian in the pseudo-spin basis as in Eq.
(1) with coefficients The model takes different phase diagrams and critical conditions based on the number of neighbors as shown in Fig. 4a-d. In this case, the quasi-energy dispersion and curvature function are real quantities given by, At criticality, the energy dispersion vanishes and curvature function becomes non-analytic.

Non-Hermitian Su-Schrieffer-Heeger chain.
Here we construct non-Hermitian SSH chain by introducing an imbalance through the intracell hopping parameter. The Hamiltonian can be written as 75,77 , where δ is the imbalance term. With the limit l → ∞ (infinite neighbors) the model becomes long-range and as α → ∞ the model reduces to short-range (where the system contains only W = 1 and W = 0 phases). The term www.nature.com/scientificreports/ t ′ l α represents extended-range intercell coupling with α as the decay parameter as schematically represented in Fig. 1b. After Fourier transformation, the Hamiltonian can be written in spin basis as in Eq. (1), with coefficients, respectively. Due to imbalance in the intracell hopping, the energy dispersion becomes complex and thus the non-Hermiticity is introduced into the model. For the current model, energy dispersion is given by The curvature function around exceptional points (0,±χ im y (k )) are given by whose integral over the closed interval gives the WN. For the non-Hermitian case, WN is calculated as the average of individual WNs around the exceptional points 75 , i.e., W = W 1 +W 2 2 . The model exhibits a interesting property called non-Hermitian skin effect, due to which the phase diagram, winding number and localized modes show the sensitivity towards the choice of boundary conditions. Initially we present the geometric properties in the periodic boundary conditions and skin effect in the later sections.

Results
In the section we study the mixed state behavior of topological models with different range of couplings. Here we take the study of pseudo-spin vectors and geometric phase for this purpose. The pseudo-spin (winding) vectors play an important role in defining the topology of a two level system. The number of time WVs rotate or wrap the center of parameter space gives the WN 42,81 . Here we consider the normalized WVs, which wrap the center of the maximally mixed states. For topological condition, WVs wrap equal to that of WN and for non topological they keeps bouncing at one side of the parameter space. For the phase transition condition, the WVs show a discontinuity in their flow (3D representation) or curvature line touches the center of the parameter space (2D representation). It is interesting that, at criticality, the angle tan θ = χ y χ x becomes indeterminate (0/0) as both χ x , χ y → 0 resulting in an ill-defined topological invariant. This situation is similar in case of non-Hermitian systems, where the azimuthal angle is complex. The modified angles around the exceptional points can be written The situation is little different in long-range interaction, where the WVs take the form of polylogarithmic functions. Here tan θ = χ y χ x becomes 0/0 at k = π(∀α) and k = 0(α > 1) respectively. This is because, at k = 0 , the function Li α (1) ∝ Ŵ(α − 1) and remain divergent. Hence, for α < 1 the WVs show a discontinuity around k = 0 , resulting in the formation of removable singularities, whose integration gives the fractional WN.
Here we consider three different ranges to understand the interplay of neighboring coupling and mixed state behavior in topological (Hermitian/non-Hermitian) systems.
Short-range couplings. If the coupling parameter has a strength only up to first neighbor, then the model can be called as short-range model, i.e., the decay parameter α → ∞ . Here, we study the mixed state behavior of Hermitian and non-Hermitian SSH chain with short-range coupling.
Hermitian case. In this case, the Hamiltonian is given by Eq. (11) with α → ∞ , whose Fourier transform yield Eq. (12) in the form The criticality occurs at t = t ′ (k = 0) and t = −t ′ (k = π) by separating W = 0 and W = 1 phases as shown in Fig. 2a. The corresponding Gibb's states are given by At T = 0 , the WVs wrap the origin of the parameter space only one time for W = 1 phases (Fig. 3a1), where as for W = 0 , the WVs do not wrap the origin (Fig. 3a2). With the introduction of the arbitrary temperature ( T = 0.1 ), the WVs show some modifications but the nature remains same as previous. The further increase in (15)  www.nature.com/scientificreports/ temperature, the curves gradually move towards one corner of the parameter space which may finally result in breakdown of topological properties. On the other hand, the Uhlmann phase become equal to Berry phase in the pure state limit (T → 0) ( Table 1). Our results are consistent with the previous work Ref. 31 .
Non-Hermitian case. In this case, the Hamiltonian is given by Eq. (14) with the limit α → ∞ , whose Fourier transform gives Eq. (15) in the form  www.nature.com/scientificreports/ The criticality condition is given by t = −t ′ ± δ(k = 0) and t = t ′ ± δ(k = π) which separate phases W = 0, 1/2 and W = 1 as shown in Fig. 2b. Instead of single gap closing point, we get two exceptional points which also produce two parameter spaces for the WVs. The combined effect in the parameter spaces decides the topological behavior of the system. The EWVs are given by, and corresponding Gibb's states are given by At T = 0 , the EWVs encircle the origin once in both parameter space for W = 1 case (Fig. 3b1,b2). For W = 1/2 , the EWVs encircle the origin of only one parameter space as shown in Fig. 3b3,b4. With the introduction of arbitrary temperature, the curve starts shifting towards one end of the parameter space. The higher temperature may destroy the topological properties of the system. The Uhlmann phase is given by Eq. (10), which becomes equal to Berry phase in the pure state limit T → 0 (Table 1).
Extended-range couplings. When the coupling strength is more than one nearest neighbor, the model is called extended model 44 and the phase diagram varies based on the number of coupling neighbors as shown in Figs. 4a-d and 6a-d. With the increase of neighbors, the possible WN also increases up to certain level and the model reduces to short-range limit with increasing value of decay parameter. This creates a staircase of topological transitions and we can observe a pattern of transitions among even-even and odd-odd WNs, based on the number of interacting neighbors 42 .
To understand the mixed state behavior, we consider the simple case r = 2 , where there are only two neighbor couplings for Hermitian and non-Hermitian situations.
Hermitian case. In this case, model Hamiltonian is given by Eq. (11) with r = 2 , whose Fourier transform is gives Eq. (12) in the form The criticality condition is given by which separate W = 0, 1 and W = 2 phases. The phase diagram is given by Fig. 4c. The Gibb's states corresponding to above parameter space are given by,  (4) and (10) respectively. Here the geometric phases are quantized in the units of π with t ′ = 1.

Region
Berry phase Uhlmann phase www.nature.com/scientificreports/ At T = 0 , the WVs encircle the origin two (one) times for W = 2(W = 1) phases as shown in Fig. 5a,b. With the introduction of an arbitrary temperature, the enclosed area shrinks but the nature remains same. The further increase of the temperature results in the localization of curves towards a corner of the parameter space which results in the breakdown of the topological properties. The Uhlmann phase is given by Eq. (10), which shows a different behavior for extended-range coupling. The Uhlmann phase recognizes W = 0 and W = 1 , but fails to recognize W = 2 . This result is also holds same to other extended-range models, with finite number of neighbor Here the star symbol represents the region of higher WNs, which are less stable compared to their lower WNs. www.nature.com/scientificreports/ couplings. This is because, the Uhlmann phase fails to recognize multiple winding of curves around the center of parameter space. The results are presented in Table 2.
Non-Hermitian case. In this case, model Hamiltonian is given by Eq. (14) with r = 2 , whose Fourier transform is gives Eq. (15) in the form The criticality occurs at which separates W = 0, 1/2, 1, 3, 2, 2, 1 and 0 topological phases as shown in Fig. 6c. The EWVs are given by The Gibb's states corresponding to above parameter space are given by, In non-Hermitian extended models, the topological properties are determined by the average behavior of parameter space corresponding to F1(k, M) and F2(k, M) . If the EWVs encircle the origin equal (unequal) number of times, results in integer (fractional) WNs. Sometimes the EWVs encircle one of the origin even (odd) number times and other origin zero times, resulting in integer (fractional) WNs. For W = 2 case, the EWVs encircle the center of each parameter space two times (Fig. 7a3,b3), while they encircle each centers only once for (26) (4) and (10) respectively.
Here the geometric phases are quantized in the units of π with t ′ = 1.

Region
Berry phase Uhlmann phase www.nature.com/scientificreports/ W = 1 . For W = 1/2 case, the EWVs encircle one of the center once and do not encircle the other (Fig. 7a1,b1). For W = 3/2 case, the EWVs encircle one of the origin twice and the other only once (Fig. 7a2,b2). In some special cases, EWVs encircle the origin of one parameter space twice and do not encircle the other, which also result in W = 1 (Fig. 7a4,b4). In non-Hermitian cases, the Uhlmann phase is given by the average of geometric  www.nature.com/scientificreports/ phase with respect to parameter spaces F1(k, M) and F2(k, M) , given by W U = W U1+W U2 2 in the pure state limit ( T → 0 ). The Uhlmann phase recognizes W = 0, 1/2 and W = 1 phases, but fails to recognize W = 3/2 and W = 2 . This is due to the limitation of the geometry, that Uhlmann, approach do not recognizes the multiple number of windings around the origin. For the same reason, Uhlmann phase fails to recognize the above mentioned special cases, even though the resulting phase is W = 1 . A detailed study is given in Table 2. These results also holds for the extended-range models with higher number of neighboring couplings.

Long-range couplings. With infinite number of coupling neighbors,the model becomes long-range and
the pseudo-spin vectors are expressed in terms of polylogarithmic function. Expansions of polylogarithmic functions gives 82 , where Ŵ function becomes ill-defined for the region α < 1 around k = 0 . This creates a removable singularity in the parameter space, where the winding vectors covers only a half rotation around the axis and bounce back to the original configuration.Thus, they fail to give a integer winding number but produces fractional winding numbers.
Hermitian case. In this case, Hamiltonian is given by Eq. (11) with l → ∞ , whose Fourier transform gives Eq. (12) in the form where the term Li α gives the polylogarithmic function 82 , which is the consequence of long-range effect. Here the criticality occurs at as shown in Fig. 4d. The Gibb's states corresponding to above parameter space are given by, Due to long-range effect, the WVs behave as sin(k) instead of sin(nk) . Thus we obtain only two topological regions with W = 0 and W = 1 . For W = 1 the WVs encircle the axis once and for W = 0 they do not encircle the axis. Due to polylogarithmic nature, Li α (1) show a discontinuous region for α < 1 , which results in an illdefined topological region for α < 1 42,44,50 . Here we observe a discontinuity at k = 0 , which acts as a removable singularity and yields fractional WN ( W = 1/2 ) (Fig. 8a1). For 1 < α < 2 , we observe a less populated WVs around k = 0 , which do not alters the resulting geometric phase (Fig. 8a2). For the range α > 2 , we observe a homogeneous distribution of WVs, which is equivalent to the short-range limit, i.e., W = 1 (Fig. 8a3). With the introduction of an arbitrary temperature, we observe a variation in density of WVs as shown in Fig. 8b1-b3. In the pure state limit ( T → 0 ), Uhlmann phase fails to recognize the fractional WN but it recognizes the integer WN as shown in Table 3.
Non-Hermitian case. In this case, Hamiltonian is given by Eq. (14) with l → ∞ , whose Fourier transform gives Eq. (15) in the form The criticality occurs at as shown in phase diagram Fig. 6d. The EWVs are given by The Gibb's states corresponding to above parameter space are given by, www.nature.com/scientificreports/   (10) respectively. Here the geometric phases are quantized in the units of π with t ′ = 1 . The top and bottom tables represents the Hermitian and non-Hermitian SSH chains respectively. www.nature.com/scientificreports/ Here we can observe two kinds of fractional WNs. For α < 1 we find fractional WN as a result of polylogarithmic nature, (which are topologically ill-defined) and for α > 1 as a result of non-Hermiticity. At T = 0 , the EWVs corresponding to the parameter space corresponding to F1(k, M) and F2(k, M) show a discontinuity for α < 1 region, resulting in an ill defined topological invariant (Fig. 9a1,b1). But this this acts as a removable singularity and yield W = 1/4 for α < 1 region. For 1 < α < 2 , the EWVs encircle the axis once with a less populated arrows around k = 0 (Fig. 9a2,b2). However, this does not influences the geometric phase of the system, and we obtain W = 1 for δ − Li α (1) < t < −δ − Li α (−1) region. For α > 2 , we get the short-range limit with W = 1 where the EWVs encircle the axis of both the parameter space (Fig. 9a3,b3). For the region ∓δ − Li α (∓1) < t < ±δ − Li α (±1) , we get W = 1/2 where the EWVs encircle the axis of any one of the parameter space (Fig. 9 a4,b4). With the introduction of arbitrary temperature, the EWVs show the modifications as shown in Fig. 9 (Fig. 9c1-c4,d1-d4). In the pure state limit T → 0 , the Uhlmann phase recognizes the integer and fractional phases for α > 1 but fails to recognize the phases in the region α < 1 . A detailed study is given in Table 3.

Region Berry phase Uhlmann phase
From the above study, we understand the limitation of Uhlmann phase in explaining the topological invariants at finite temperature, especially for extended-range models. In extended-range (Hermitian and non-Hermitian) models, the Uhlmann phase fails to reproduce the phase diagram at pure state limit. Here we use another important approach, interferometric geometric phase to explain the finite temperature behavior of extended-range topological models.

Interferometric geometric phase.
Here a different geometric phase for mixed states is introduced through the concept of interferometer 83 . The purification of normalized of state |ω� ∈ H ω is with H ω = H S ⊗ H A and |ψ i � ∈ H A . Here H S is the Hilbert space of the system and H A is the Hilbert space spanned by the ancillary states with i dimensions. By tracing over the ancillary states, we write the density matrix as, ρ = Tr A (|ω��ω|) . By parameterizing the density operators with a continuous parameter k, such that the eigenvalues of ρ(k) for each k yield non-degenerate values, www.nature.com/scientificreports/ The eigenstates evolve in parallel manner such that two infinitesimally separated eigenstates in H s show parallel transport in order to fix the phase ambiguity of |ω� through gauge fixing. The parallelism is given by which yield the interferometric phase of ρ(k) across the Brillouin zone. i.e., . It is to be noted that, the state |ψ k � is affected by parallel transport, while the ancillary states |ψ k ′ � are not. Hence, the interferometric phase reduces to Berry phase only if ρ(k) is a parameterized density operator of pure states 31,60 .
Thus, we can observe the higher WN as well as fractional WNs in the pure state limit. This geometric phase overcomes the limitations of Uhlmann phase and can be effectively used as topological measure in Hermitian and non-Hermitian systems. For the Hermitian system, the method remains straightforward. For the non-Hermitian systems, the interferometric phase can be calculated for the individual parameter spaces corresponding to F 1,2 (k, M) and combined effect can be observed through the average W I1 +W I2 2 . Thus, we understand that interferometric geometric phase is a better tool to study the mixed state behavior of extended-range (Hermitian and non-Hermitian) topological models. A comparison is given in Fig. 10.
Topology at gapless condition. Localization is an important property which has direct relation with the topological invariant. We can observe a one to correspondence between the number of localized modes and neighbor coupling up to some extent 44 . However, increase in the number of neighbor coupling result in the generation of higher winding numbers. With the increase of decay parameter, we observe a staircase of topological transitions among corresponding even-even and odd-odd winding numbers 42 . The bulk phases contain localized modes, protected by certain discrete symmetries. This naturally exhibits bulk-boundary correspondence and it is believed that, bulk gap is necessary to exhibit bulk-boundary correspondence. Recently, there have been observations of localization even at criticality exhibiting bulk-boundary correspondence, signaling the bulk-boundary correspondence even in the absence of bulk gap 72,73,84 . By definition, topological invariant is ill-defined at criticality, but there are efforts to define topology at criticality through different means. Here we consider a case of localization at criticality and define the topological invariant by excluding the infinitesimal neighborhood of singular points in the Brillouin zone 64 . www.nature.com/scientificreports/ where {k i } is the set of critical points in the momentum space. For the Hermitian case, we find a unique behavior at k = π criticality. The critical line µ = −t ′ (−1 + 1 2 α ) line contains a multi-critical point at α = 1 , where the critical line α < 1 witness the transition W : 2 → 1 and α > 1 witnesses W : 0 → 1 respectively (Fig. 4c). During the transition W : 2 → 1 , out of two edge modes, one localizes at criticality and the other transfers to the W = 1 gapped phase. Thus, on the same critical line, we observe a localized mode for α < 1 and non localization for α > 1 , which creates a topological transition along criticality across a multi-critical point 72,73,84 .
For the region α < 1 , the WVs show two loops, with the inner one touching the center of the parameter space (Fig. 11 a), while for α > 1 , WVs show a single loop touching the center (Fig. 11b). The change in configuration occurs at α = 1 as shown in Fig. 11c. Even though, all the configurations are occurring on the line t = −t ′ (−1 + 1 2 α ) , the topological behavior of them are different. Due to the localization property, we observe different WNs even at criticality. But at a finite temperature, Uhlmann phase does not recognizes the WNs at criticality. This limitation can be overcome by using interferometric phase, where we can recognize the WNs at criticality. The comparison of geometric phases are given in Table 4.

Non-Hermitian skin effect
So far, we have constructed the phase diagram and calculated the topological invariant by assuming the periodic boundary condition to Hermitian and non-Hermitian SSH models. Under general conditions, topological invariant is calculated using Bloch band structure and is in good agreement with the number of localized edge modes. This scenario is true in Hermitian systems, while non-Hermitian systems behave differently. Non-Hermitian systems are sensitive to the boundary conditions, which produce different phase diagram for periodic and open boundary conditions. This behavior is due to the phenomenon called non-Hermitian skin effect, which signals the necessity of the construction of 'non-Bloch topological invariant' . Due to non-Hermitian skin effect, the topologically protected edge modes depends on the spacial dimension and the symmetry class rather than the topological invariant 34 Table 4. A comparison of different pure and mixed state geometric phases of Hermitian SSH chain witnessing the topological transition along the critical line t = −t ′ (−1 + 1 2 α ) . Here the geometric phases are quantized in the units of π with t ′ = 1 . The Berry phase, Uhlmann and interferometric phases are calculated through Eqs. (4), (10) and (41) respectively.

Region
Berry phase Uhlmann phase Interferometric phase www.nature.com/scientificreports/ with k ∈ [0, 2π) . The presence of sub lattice symmetry creates Z topological phase in the presence of line gap complex energy spectrum. The topological invariant has the similar form of that of Hermitian conditions. For the region with point gap complex energy spectrum, there exists Z ⊕ Z topological phases, and invariant is given by, The W P becomes zero for an integer W L , whereas W P becomes integer value for fractional W L .
Here we present the geometric phases of non-Hermitian SSH chain with different range of couplings (Fig. 12), which is different than its counterpart in periodic boundary condition (Fig. 12a1,a2). It clearly shows the sensitivity of phase diagram, towards the choice of boundary conditions. In generalized Brillouin zone, there can occur a significant calculation errors in defining the topological invariant for extended-range due to various reasons 89,90 . In addition, the Uhlmann phase shows a limitation in defining the geometric phase at finite temperature. The higher winding numbers are not defined by the Uhlmann phase and the phase boundary does not shows an exact point of transition (Fig. 12b1,b2). Instead, the phase is not exactly integer quantized (at least in extended-range) add the phase boundary is spreaded. On the other hand the interferometric phase shows the similar behavior that of Berry phase in the finite temperature ( T → 0 ) limit (Fig. 12c1,c2). However, so far the physical interpretation of Uhlmann phase has not been revealed in an extensive way, which creates a difficulty in interpreting the physical meaning of geometric phases at finite temperature. The Uhlmann phase also has a limitation in establishing the relation between the geometry at finite temperature and the bulk-edge correspondence 23,31 . The Uhlmann phase works on the principle of the parallel transport of density matrices at finite temperature, thus it recognized the critical temperature and the topological region, while it fails to determine the number of edge modes due to the memory effect 31,60 . On the other hand, the interferometric phase works straightforwardly based on the principle of ancillary states unlike Uhlmann phase (which works on operator method), and behaves similar to the Berry phase at T → 0 limit. However, one can predict the criticality, critical temperature and geometric phases of mixed state topological systems through different means (both for periodic and open boundary conditions), but the understanding towards the bulk-edge correspondence needs some more study.

Discussion
Mixed state behavior of quantum system is an efficient way to understand the thermal fluctuations, especially to define the topology at finite temperature ( T → 0 ). There are a few studies in literature to understand the finite temperature limit of topological system with short-range and long-range interactions. Here we made an attempt to understand the behavior of extended topological models in finite temperature limit. We have extended our  observations to the non-Hermitian topological system to understand the geometry and geometric phases at finite temperature. Introduction of extended range interaction in a topological system is an efficient way to generate the higher WNs, which is an static way. The similar effect can be done by the quenching/periodic driving, which is a dynamical method. In an equilibrium topological system, generally the WN is in correspondence with the localized edge modes in the gapped phases (at least up to some range r < L/2 . Sometimes this may not be true if certain symmetries like time reversal is broken). Here we made an effort to study the extended topological chains in the finite temperature limit. We have used the mixed geometric phases like Uhlmann and interferometric phases to understand the topology at finite temperature. We have found that Uhlmann approach has a limitation to define the higher WNs and this can be verified by defining the winding vectors in the form of Gibb's state. We use interferometric geometric phase to find the geometric phase at finite temperature as shown in Fig. 10. Longrange topological models are the platform to realize the massive edge modes (along with Majorana zero modes), which can also effectively used as topological qubit. Here we obtain fractional WNs for the region where massive edge modes dominate. We carry out finite temperature and observe that the Uhlmann phase has a limitation in defining the fractional WNs. The authors of Ref. 60 have worked on similar model and mentioned a different undefined region for Uhlmann phase, due to the nature of superconducting pairing term.
Non-Hermiticity is a prominent area of quantum mechanics and here we have adopted biorthonormal basis vectors to define the topology of the models. The concept can be extended to the mixed states and hence it is possible to understand the geometric phases of non-Hermitian models at finite temperature. However, the non-Hermitian topological models are quite different than the Hermitian systems and have a separate set of periodic table of symmetry protected classes. Moreover, they are sensitive to the boundary condition, and phase diagram can differ according to that. Here we have analyzed chiral non-Hermitian SSH chain and their topology at finite temperature. We observe that the Uhlmann phase recognizes the fractional and integer phases in the short-range limit, but fails to recognize the same in extended-range. It is also to be noted that the similar situation has been observed in the open boundary condition (in the presence of non-Hermitian skin effect), where the Uhlmann phase shows a limitation in defining the higher winding numbers.
As per the traditional definition of topology, WN can be well defined in the gapped phases and ill-defined at the gapless regions. But the experimental and theoretical study at criticality strengthens the possibility of localized states even at criticality. This creates a need of defining topology at criticality and we adopted modified definition to address this issue. The study of Uhlmann phase shows a limitation in defining topological invariant at criticality and fails to recognize the transition among gapless regions across a multi-critical point.
To conclude, we have analyzed the mixed state behavior of Hermitian and non-Hermitian topological models and tried to calculate the geometric phase at T → 0 limit. Among the study of geometric phase for mixed states, Uhlmann phase has showed a limitation in defining topological invariant for extended-range at finite temperature. This limitation can be overcome by the interferometric phase in the pure state limit. We also extend our study to non-Hermitian models and analyze the limitation of Uhlmann phase in defining the topological invariants for extended-range models. We have analyzed the geometric phases at open boundary condition and tried to understand the behavior of geometric phases in the presence of non-Hermitian skin effect. We understand that the physics of Uhlmann phase is complicated and there needs further study in this regard. We agree with the statement of Viyuela et al. 29 , and Anderson et al. 31 that the Uhlmann phase does not determine the fate of edge modes at finite temperature' and we extend validation of this statement to the extended range interaction of Hermitian and non-Hermitian models. We also find that interferometric phase can serve as an efficient tool calculate the topology of (Hermitian and non-Hermitian) extended-range models.

Method
Derivation of effective winding number for non-Hermitian systems. The generalized expression for WN is given by the Eq. (4), with periodic boundary condition the expression is Due non-Hermiticity, the Eigen vectors of the Hamiltonian can be written as 75 where The non-Hermiticity induces the complexity in winding vectors(at in least one component) with the criticality condition χ 2 x + hχ 2 y = 0 . Here we observe exceptional points (at least two), instead of a single Dirac cone (band gap closing point), whose position can be understood by the complex analysis as, www.nature.com/scientificreports/ As our model contains imaginary component only in χ y term, the exceptional points are located at (0, χ im y ) and (0, −χ im y ) under the criticality condition χ 2 x + χ 2 y = 0 . Due to the non-Hermiticity, each exceptional point induces its own origin of pseudo-spin space and corresponding winding vectors. Thus we work on extended winding vectors to understand the non-Hermitian effect in the parameter space. i.e., which correspond to parameter space F 1 (k, M) and F 2 (k, M) respectively (Fig. 13).
As a result of non-Hermiticity the azimuthal angle becomes complex, i.e., φ = φ re + iφ im , where the real and imaginary parts contribute to the argument and amplitude respectively. Thus the complex angle can be calculated as, where tan(φ 1 ) = χ re y (k)+χ im x (k) χ re x (k)+χ im y (k) and tan(φ 2 ) = χ re y (k)+χ im x (k) χ re x (k)−χ im y (k) . Thus azimuthal angle φ 1 (φ 2 ) corresponding to first (second) exceptional point can be expressed in real angles. The integration of curvature functions F1(k, M), F2(k, M) corresponding these azimuthal angles give the WN as i.e., Encircling of these extended winding vectors around the centers of both the parameter space (equal time) yields integer WN. If the encircling is only around one of them yields fractional and neither of them yields W = 0 respectively. (48) χ re x (k) = −χ im y (k)and χ re y (k) = χ im x (k) or χ re x (k) = χ im y (k)and χ re y (k) = −χ im x (k) (49) χ x (extended) =χ re x (k) ± χ im y (k), χ y (extended) =χ re y (k),  www.nature.com/scientificreports/ Generalized Brillouin zone and topological invariant. Due to the existence of non-Hermitian skin effect, the topological phase diagram becomes sensitive to boundary conditions. Here we adopt the non-Block band theory to treat the topological models under open boundary conditions 85 . For a short-range model (Eq. (14) with α → ∞ ), the real space Eigen equation leads Considering the ansatz |ψ� = j |φ (j) � , where each |φ (j) � takes the form (omitting the j index temporarily) (φ n,A , φ n,B ) = β n (φ A , φ B ) . We obtain the relation the energy equation is given by Under the limit E → 0 , we obtain the roots Restoring the j index, we obtain Thus the general solution for ψ can be written as linear superposition of φ as ψ = n β (n) φ . For the spectrum in thermodynamic limit, the bulk-Eigenstates follow the rule |β 1 | = |β 2 | , which leads to For the case r = 1 , the non-Hermitian model behaves similar to that of the Hermitian counterpart 88 . Topological invariant Due to non-Bloch band structure, the components get modified as e ik → β, e −ik → β −1 and the Hamiltonian is given by where σ ± = (σ x +iσ y ) 2 . Here the k term takes the complex value k → k − i ln(r 0 ) . The energy relations are given by where |ũ R � = σ z |u R �, |ũ L � = σ z |u L � which follow the condition �u L |u R � = �ũ L |ũ R � = 1, �u L |ũ R � = �ũ L |u R � = 0 leading to the Q-matrix Q(β) = |ũ R (β)��ũ L (β)| − |u R (β)��u L (β)| . The topological invariant in generalized Brillouin zone is given by where H(k) can be written in the corrected form as which defines the topological invariant in the generalized Brillouin zone 10,85-88 . Extended-range couplings For the extended-range models such as r = 2 , the real space wavefunctions can be written as, www.nature.com/scientificreports/ Similar methodology can be adopted to define the geometric phase for open boundary condition. Here the term B (Eq. 69) behaves independent of periodicity, which signals the generalization of Uhlmann phase to incorporate non-Hermitian skin effect. By replacing the Block wave function u k with non-Block function β , this can be achieved. The parallelism of the density matrix allows the wavefunctions to define Uhlmann phase at finite temperature. The 'Q' matrix defines the parallel condition and the trace of the matrix defined the geometric phase in the generalized Brillouin zone.